Dense holomorphic curves in spaces of holomorphic maps and applications to universal maps
Dense holomorphic curves in spaces of holomorphic maps and applications to universal maps
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全纯映射空间中的稠密全纯曲线及其在通用映射中的应用
DOI：
10.1142/s0129167x17500288
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发表时间：
2017
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影响因子：
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通讯作者：
Yuta Kusakabe
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作者：
Yuta Kusakabe
研究了Stein空间中的全纯映射空间中何时存在稠密全纯曲线。我们首先证明，对于任何有界凸域$\Omega\Subset\mathbb{C}^n$和任何连通复流形$Y$，空间$\mathcal{O}（\Omega，Y）$包含一个稠密的全纯圆盘。我们的第二个结果指出$Y$是Oka流形当且仅当对于任何Stein空间$X$，在$\mathcal{O}（X，Y）$的每个路径分量中存在稠密整曲线。
在本文的后半部分，我们将上述结果应用于泛函数论。证明了对任意有界凸域$\Omega\Subset\mathbb{C}^n$，$\Omega$的任意不动点自由自同构和任意连通复流形Y$，存在一个泛映射$\Omega\to Y$.我们还通过泛映射的存在性刻画了Oka流形。
We study when there exists a dense holomorphic curve in a space of holomorphic maps from a Stein space. We first show that for any bounded convex domain $\Omega\Subset\mathbb{C}^n$ and any connected complex manifold $Y$, the space $\mathcal{O}(\Omega,Y)$ contains a dense holomorphic disc. Our second result states that $Y$ is an Oka manifold if and only if for any Stein space $X$ there exists a dense entire curve in every path component of $\mathcal{O}(X,Y)$.
In the second half of this paper, we apply the above results to the theory of universal functions. It is proved that for any bounded convex domain $\Omega\Subset\mathbb{C}^n$, any fixed-point-free automorphism of $\Omega$ and any connected complex manifold $Y$, there exists a universal map $\Omega\to Y$. We also characterize Oka manifolds by the existence of universal maps.